Page 153 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

144 Angular momentum

For m equal to ÿl ‡ 1, equation (5.38a) gives
1 ^ 1 ^ 2
Y l,ÿl‡2 ˆ p  L ‡ Y l,ÿl‡1 ˆ p  L Y l,ÿl
‡
2(2l ÿ 1) " 2(2l)(2l ÿ 1) " 2
where equation (5.46) has been introduced in the last term. If we continue in
the same pattern, we ®nd
1 ^ 1 ^ 3
Y l,ÿl‡3 ˆ p  L ‡ Y l,ÿl‡2 ˆ p  L Y l,ÿl
:
‡
3(2l ÿ 2) " 2 3(2l)(2l ÿ 1)(2l ÿ 2) " 3
.
.
.
s 
(2l ÿ k)! 1 k
Y l,ÿl‡k ˆ ^
L Y l,ÿl
k!(2l)! " k ‡
where k is the number of steps in this sequence. We now set k ˆ l ‡ m in the
last expression to obtain
s 
(l ÿ m)! 1 l‡m
L
Y lm ˆ ^ Y l,ÿl (5:47)
(l ‡ m)!(2l)! " l‡m ‡
If the number of steps k is less than the value of l, then the integer m is
negative; if k equals l, then m is zero; if k is greater than l, then m is positive;
and ®nally if k equals 2l, then m equals its largest value of l.
^ l‡m
The next step in this derivation is the evaluation of L Y l,ÿl using equation
‡
^
(5.37a). If the operator L ‡ in (5.37a) acts on Y l,ÿl (è, j) as given in (5.45), we
have

^ ij @ ‡ i cot è @ sin è e ÿilj
l
L ‡ Y l,ÿl ˆ c l "e
@è @ö

l
ˆ c l "e ÿi(lÿ1)j d ‡ l cot è sin è
dè
d sin è
2l
ˆ c l "e ÿi(lÿ1)j ‡ l cot è
dè sin è
l

1 d
2l
ˆ c l "e ÿi(lÿ1)j sin è ÿ l sin lÿ1 è cos è ‡ l sin lÿ1 è cos è
sin è dè
l
1 d
2l
ˆÿc l "e ÿi(lÿ1)j sin è
sin lÿ1 è d(cos è)
where for brevity we have de®ned c l as
r 
1 (2l ‡ 1)!
c l ˆ (5:48)
2 l! 4ð
l